Study on Efficient solution of Multi-objective Nonlinear Programming problem Involving Semilocally Pseudolinear functions

Abstract

Isermann (1978) established that every efficient solution of optimization problem is properly efficient if the objective functions are linear and feasible set is polyhedral. Choo (1984) proved this result when the objective functions are linear fractional and the feasible set is a compact polyhedron. Chew and Choo (1984) established that every efficient solution of optimization problem which satisfies the following boundedness condition is properly efficient if all the objective functions and constraint functions are pseudolinear. A feasible point χ∈Χ is said to satisfy the boundedness condition if the set {Pi(χ*,χ)Pj(χ*,χ):fi(χ)<fi(χ*),fj(χ)>fj(χ*),1≤i,j≤κ} is bounded above, where pi is a proportional function corresponding to fi,i = 1,2,3,………κ. Gulati and Islam (1990) have shown that the above result still holds if we take the objective functions to be pseudolinear, constraint functions to be quasiconvex and certain constraint qualification is satisfied. In this paper we show that every efficient solution that satisfies the above boundedness condition and generalized constraint qualification, is properly efficient if all the objective functions and constraint functions defined on a convex set are semilocally pseudolinear and their right differentials at that efficient solution are convex.